Solving $\int_0^{\infty}\!\sqrt{x}\,J_1(a\sqrt{x})\cos(\omega x)\,dx$ and $\int_0^{\infty}\!\sqrt{x}\,J_1(a\sqrt{x})\sin(\omega x)\,dx$

Question

I should solve the following integrals:

$$\int_0^{\infty}\!\sqrt{x}\,J_1(a\sqrt{x})\cos(\omega x)\,dx$$

and

$$\int_0^{\infty}\!\sqrt{x}\,J_1(a\sqrt{x})\sin(\omega x)\,dx$$

with $a, \omega >0$.

I cannot find them in Gradshteyn & Ryzhik.

Any suggestions? Thanks

Answer 1

Even the special case $a=1,\omega=0$ $$ \int _{0}^{s}\!\sqrt {x}\;{{\rm J}_1\left(\sqrt {x}\right)}{dx}=-2\,s\; {{\rm J}_0\left(\sqrt {s}\right)}+4\, {{\rm J}_1\left(\sqrt {s}\right)}\sqrt {s} $$ has limsup $+\infty$ and liminf $-\infty$ as $s \to \infty$.

So why do you think the case $\omega>0$ converges?